Finding The Linear Function For Basketball Ticket Costs A Step By Step Guide
Introduction
In the realm of mathematics, particularly in the study of linear functions, real-world scenarios often provide a practical context for understanding abstract concepts. This article delves into a common situation – the cost of ordering tickets to a basketball game online – and demonstrates how a linear function can be used to model the total cost based on the number of tickets purchased. We will explore the given information, which includes a fixed service fee and the total cost for a specific number of tickets, and then derive the linear function that accurately represents the relationship between the number of tickets ordered and the overall cost. Understanding such linear functions is crucial not only in mathematics but also in everyday decision-making, as they allow us to predict costs, analyze trends, and make informed choices. In this specific case, we aim to determine the linear function that describes the total cost, denoted as 'c', when 'x' tickets are ordered, providing a clear and concise formula for calculating expenses related to purchasing basketball game tickets online.
Problem Statement: Deconstructing the Cost Equation
Let's dissect the problem at hand. Tickets for a basketball game are available for online purchase, with each ticket having a set price. In addition to the cost per ticket, a fixed service fee of $5.50 is applied to each order, regardless of the number of tickets purchased. We are given a crucial data point: the total cost for ordering 5 tickets is $108.00. Our objective is to determine the linear function that represents 'c', the total cost, as a function of 'x', the number of tickets ordered. This task involves several key steps. First, we need to identify the components of the cost: the fixed service fee and the variable cost based on the number of tickets. Next, we can use the provided information – the total cost for 5 tickets – to calculate the cost per ticket. Once we know the cost per ticket and the service fee, we can construct the linear function in the form of c = mx + b, where 'm' represents the slope (the cost per ticket) and 'b' represents the y-intercept (the service fee). Finally, we will have a linear function that allows us to calculate the total cost for any number of tickets ordered, providing a practical tool for budgeting and planning.
Setting Up the Linear Equation: Variables and Constants
To effectively model the cost of basketball tickets, we begin by defining our variables and constants. Let 'c' represent the total cost in dollars for ordering tickets online. This is the dependent variable, as it depends on the number of tickets ordered. Let 'x' represent the number of tickets ordered, which is the independent variable. The problem states that there is a fixed service fee of $5.50 applied to each order. This is a constant value that does not change regardless of the number of tickets purchased. Let's denote the cost per ticket as 'p'. This is another constant that needs to be determined. The total cost 'c' can be expressed as the sum of the cost of the tickets (the number of tickets multiplied by the cost per ticket) and the service fee. Mathematically, this can be written as c = px + 5.50. This equation represents a linear function in slope-intercept form, where 'p' is the slope and 5.50 is the y-intercept. Our next step is to use the given information – the total cost for 5 tickets – to solve for 'p', the cost per ticket. This will allow us to complete the linear function and have a fully defined equation that represents the relationship between the number of tickets ordered and the total cost.
Calculating the Cost Per Ticket: Solving for the Slope
Now, let's leverage the information provided to calculate the cost per ticket. We know that when 5 tickets are ordered (x = 5), the total cost is $108.00 (c = 108). We can substitute these values into our equation, c = px + 5.50, to solve for 'p'. Substituting the values, we get 108 = 5p + 5.50. To isolate 'p', we first subtract 5.50 from both sides of the equation: 108 - 5.50 = 5p, which simplifies to 102.50 = 5p. Next, we divide both sides by 5 to solve for 'p': 102.50 / 5 = p. This gives us p = 20.50. Therefore, the cost per ticket is $20.50. This value represents the slope of our linear function, indicating the rate at which the total cost increases with each additional ticket purchased. Now that we have calculated the cost per ticket, we have all the necessary components to construct the complete linear function that represents the total cost of ordering basketball tickets online. This equation will allow us to easily calculate the total cost for any number of tickets.
Constructing the Linear Function: Putting It All Together
With the cost per ticket calculated to be $20.50, we can now construct the complete linear function that represents the total cost 'c' for ordering 'x' tickets. We have our slope, m = 20.50, and our y-intercept, b = 5.50. Plugging these values into the slope-intercept form of a linear function, c = mx + b, we get c = 20.50x + 5.50. This equation is the solution to our problem. It accurately represents the total cost of ordering basketball tickets online, taking into account both the cost per ticket and the fixed service fee. The coefficient 20.50 represents the cost per ticket, meaning that for each additional ticket ordered, the total cost increases by $20.50. The constant 5.50 represents the fixed service fee, which is added to the cost regardless of the number of tickets purchased. This linear function provides a clear and concise way to calculate the total cost for any number of tickets. For example, if someone wants to order 10 tickets, they can simply substitute x = 10 into the equation to find the total cost: c = 20.50(10) + 5.50 = 205 + 5.50 = 210.50. Therefore, the total cost for ordering 10 tickets would be $210.50.
Interpreting the Linear Function: Real-World Implications
The derived linear function, c = 20.50x + 5.50, not only provides a mathematical representation of the total cost but also offers valuable insights into the real-world implications of ordering basketball tickets online. The equation highlights the two primary cost components: the variable cost, which is directly proportional to the number of tickets ordered (20.50x), and the fixed cost, which is the service fee (5.50). This distinction is crucial for understanding how the total cost scales with the number of tickets. For instance, the equation clearly shows that the cost increases linearly with each additional ticket purchased, with the rate of increase being $20.50 per ticket. This can help individuals or groups budget their expenses when planning to attend a basketball game. Furthermore, the equation reveals the impact of the service fee. Even if only one ticket is ordered, the service fee adds $5.50 to the total cost. This information can influence purchasing decisions, as it might be more cost-effective to order multiple tickets at once to minimize the impact of the service fee per ticket. In a broader context, this linear function exemplifies how mathematical models can be used to represent and analyze real-world scenarios, providing a powerful tool for decision-making in various aspects of life.
Conclusion: The Power of Linear Functions in Everyday Life
In conclusion, we have successfully derived the linear function c = 20.50x + 5.50, which represents the total cost 'c' for ordering 'x' basketball tickets online. This exercise demonstrates the practical application of linear functions in modeling real-world scenarios. By breaking down the problem into its components – the fixed service fee and the variable cost per ticket – we were able to construct an equation that accurately represents the relationship between the number of tickets ordered and the total cost. This linear function not only allows us to calculate the total cost for any number of tickets but also provides valuable insights into the cost structure, highlighting the impact of both the cost per ticket and the service fee. This understanding can inform purchasing decisions and facilitate budgeting. More broadly, this example underscores the power of linear functions as a mathematical tool for representing and analyzing real-world phenomena. From calculating costs to predicting trends, linear functions play a crucial role in various aspects of our daily lives, enabling us to make informed decisions and navigate the world around us with greater understanding.
